SENSITIVITY ANALYSIS FOR SYSTEM OF PARAMETRIC GENERALIZED QUASI-VARIATIONAL INCLUSIONS INVOLVING R-ACCRETIVE MAPPINGS

Title & Authors
SENSITIVITY ANALYSIS FOR SYSTEM OF PARAMETRIC GENERALIZED QUASI-VARIATIONAL INCLUSIONS INVOLVING R-ACCRETIVE MAPPINGS

Abstract
In this paper, using proximal-point mappings technique of Raccretive mappings and the property of the fixed point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of the system of parametric generalized quasi-variational inclusions involving R-accretive mappings in real uniformly smooth Banach space. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to parameters. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [3, 23, 24, 32, 33, 34].
Keywords
system of parametric generalized quasi-variational inclusions;proximal-point mappings;R-accretive mappings;strongly accretive mappings;mixed Lipschitz continuous;H-Lipschitz continuous;
Language
English
Cited by
References
1.
S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl. 201 (1996), no. 2, 609-630.

2.
R. P. Agarwal, Y.-J. Cho, and N.-J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13 (2000), no. 6, 19-24.

3.
R. P. Agarwal, N.-J. Huang, and M.-Y. Tan, Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions, Appl. Math. Lett. 17 (2004), no. 3, 345-352.

4.
J. P. Aubin and A. Cellina, Differential Inclusions: Set-valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264. Springer-Verlag, Berlin, 1984.

5.
C. Baiocchi and A. Capelo, Variational and Quasi-variational Inequalities, John Wiley & Sons, Inc., New York, 1984.

6.
A. Bensoussan and J. L. Lions, Applications of Variational Inequalities, Gauthiers-Villars, Bordas, Paris, 1984.

7.
H. Brezis, Equations et in´equations non lin´eaires dans les espaces vectoriels en dualite,Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 1, 115-175.

8.
C. E. Chidume, K. R. Kazmi, and H. Zegeye, Iterative approximation of a solution of a general variational-like inclusion in Banach spaces, Int. J. Math. Math. Sci. 2004 (2004), no. 21-24, 1159-1168.

9.
I. Cioreneseu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers Group, Dordrecht, 1990.

10.
J. Crank, Free and Moving Boundary Problems, The Clarendon Press, Oxford University Press, New York, 1984.

11.
S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res. 13 (1988), no. 3, 421-434.

12.
K. Demling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

13.
V. F. Demyanov, G. E. Stovroulakis, L. N. Poyakova, and P. D. panogiotopoulos, Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics, Nonconvex Optimization and its Applications, 10. Kluwer Academic Publishers, Dordrecht, 1996.

14.
X.-P. Ding, Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions, Appl. Math. Lett. 17 (2004), no. 2, 225-235.

15.
X.-P. Ding and C. L. Luo, On parametric generalized quasi-variational inequalities, J. Optim. Theory Appl. 100 (1999), no. 1, 195-205.

16.
Y.-P. Fang and N.-J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004), no. 6, 647-653.

17.
G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 7 (1963/1964), 91-140.

18.
G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 7 (1963/1964), 91-140.

19.
R. Glowinski, Numerical Methods For Nonlinear Variational Problems, Springer-Verlag, New York, 1984.

20.
R. Glowinski, J. L. Lions, and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland Publishing Co., Amsterdam-New York, 1981.

21.
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, Inc., New York, 1984.

22.
K. R. Kazmi and F. A. Khan, Iterative approximation of a unique solution of a system of variational-like inclusions in real q-uniformly smooth Banach spaces, Nonlinear Anal. 67 (2007), no. 3, 917-929.

23.
K. R. Kazmi and F. A. Khan, Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-$\eta$-accretive mappings, J. Math. Anal. Appl. 337 (2008), no. 2, 1198-1210.

24.
K. R. Kazmi and F. A. Khan, Sensitivity analysis for parametric general set-valued mixed variational-like inequality in uniformly smooth Banach space, Math. Inequal. Appl. 10 (2007), no. 2, 403-415.

25.
T. C. Lim, On fixed point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985), no. 2, 436-441.

26.
Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, Sensitivity analysis for parametric completely generalized nonlinear implicit quasivariational inclusions, J. Math. Anal. Appl. 277 (2003), no. 1, 142-154.

27.
J. J. Moreau,Proximiteet dualite dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273-299.

28.
R. N. Mukherjee and H. L. Verma, Sensitivity analysis of generalized variational inequalities, J. Math. Anal. Appl. 167 (1992), no. 2, 299-304.

29.
S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.

30.
M. A. Noor and K. I. Noor, Sensitivity analysis for quasi-variational inclusions, J. Math. Anal. Appl. 236 (1999), no. 2, 290-299.

31.
M. A. Noor, Sensitivity analysis framework for general quasi-variational inclusions, Comput. Math. Appl. 44 (2002), no. 8-9, 1175-1181.

32.
J. Y. Park and J. U. Jeong, Parametric generalized mixed variational inequalities, Appl. Math. Lett. 17 (2004), no. 1, 43-48.

33.
J. W. Peng and X. J. Long, Sensitivity analysis for parametric completely generalized strongly nonlinear implicit quasi-variational inclusions, Comput. Math. Appl. 50 (2005), no. 5-6, 869-880.

34.
J. W. Peng and D. Zhu, A new system of generalized mixed quasi-variational inclusions with (H,$\eta$)-monotone operators, J. Math. Anal. Appl. 327 (2007), no. 1, 175-187.

35.
S. M. Robinson, Sensitivity analysis of variational inequalities by normal-map techniques, Variational inequalities and network equilibrium problems (Erice, 1994), 257-269, Plenum, New York, 1995.

36.
G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.

37.
N. D. Yen, Holder continuity of solutions to a parametric variational inequality, Appl. Math. Optim. 31 (1995), no. 3, 245-255.